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Factorization using Difference of Squares

Factor binomials that are composed of subtracted perfect square terms

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Factorization using Difference of Squares

Factorization Using Difference of Squares 

When you learned how to multiply binomials we talked about two special products.

The sum and difference formula:(a+b)(ab)The square of a binomial formulas:  (a+b)2(ab)2=a2b2=a2+2ab+b2=a22ab+b2

In this section we’ll learn how to recognize and factor these special products.

Factor the Difference of Two Squares

We use the sum and difference formula to factor a difference of two squares. A difference of two squares is any quadratic polynomial in the form a2b2, where a and b can be variables, constants, or just about anything else. The factors of a2b2 are always (a+b)(ab); the key is figuring out what the a and b terms are.

 

Factoring the Difference of Squares

1. Factor the difference of squares:

 

a) x29

Rewrite x29 as x232. Now it is obvious that it is a difference of squares.

The difference of squares formula is:

a2b2=(a+b)(ab)

Let’s see how our problem matches with the formula:

x232=(x+3)(x3)

The answer is:

x29=(x+3)(x3)

We can check to see if this is correct by multiplying (x+3)(x3):

  x+3x33x9x2+3xx2+0x9

The answer checks out.

Note: We could factor this polynomial without recognizing it as a difference of squares. With the methods we learned in the last section we know that a quadratic polynomial factors into the product of two binomials:

(x)(x)

We need to find two numbers that multiply to -9 and add to 0 (since there is no xterm, that’s the same as if the xterm had a coefficient of 0). We can write -9 as the following products:

9=199=1(9)9=3(3)andandand1+9=81+(9)=83+(3)=0These are the correct numbers.

We can factor x29 as (x+3)(x3), which is the same answer as before. You can always factor using the methods you learned in the previous section, but recognizing special products helps you factor them faster.

b) x2100

Rewrite x2100 as x2102. This factors as (x+10)(x10).

c) x21

Rewrite x21 as x212. This factors as (x+1)(x1).

2. Factor the difference of squares:

 

a) 16x225

Rewrite 16x225 as (4x)252. This factors as (4x+5)(4x5).

b) 4x281

Rewrite 4x281 as (2x)292. This factors as (2x+9)(2x9).

c) 49x264

Rewrite 49x264 as (7x)282. This factors as (7x+8)(7x8).

3. Factor the difference of squares:

 

a) x2y2

x2y2 factors as (x+y)(xy).

b) 9x24y2

Rewrite 9x24y2 as (3x)2(2y)2. This factors as (3x+2y)(3x2y).

c) x2y21

Rewrite \begin{align*} x^2 y^2 - 1\end{align*} as \begin{align*}(xy)^2 - 1^2\end{align*}. This factors as \begin{align*}(xy + 1)(xy - 1)\end{align*}.

 

 

Examples

Factor the difference of squares:

 

Example 1

\begin{align*}x^4 - 25\end{align*}

Rewrite \begin{align*}x^4 - 25\end{align*} as \begin{align*}(x^2)^2 - 5^2\end{align*}. This factors as \begin{align*}(x^2 + 5)(x^2 - 5)\end{align*}.

Example 2

\begin{align*}16x^4 - y^2\end{align*}

Rewrite \begin{align*}16x^4 - y^2\end{align*} as \begin{align*}(4x^2)^2 - y^2\end{align*}. This factors as \begin{align*}(4x^2 + y)(4x^2 - y)\end{align*}.

Example 3

\begin{align*}x^2 y^8 - 64z^2\end{align*}

Rewrite \begin{align*}x^2 y^4 - 64z^2\end{align*} as \begin{align*}(xy^2)^2 - (8z)^2\end{align*}. This factors as \begin{align*}(xy^2 + 8z)(xy^2 - 8z)\end{align*}.

Review

Factor the following differences of squares.

  1. \begin{align*}x^2 - 4\end{align*}
  2. \begin{align*}x^2 - 36\end{align*}
  3. \begin{align*}-x^2 + 100\end{align*}
  4. \begin{align*}x^2 -400\end{align*}
  5. \begin{align*}9x^2 - 4\end{align*}
  6. \begin{align*}25x^2 - 49\end{align*}
  7. \begin{align*}9a^2 - 25b^2\end{align*}
  8. \begin{align*}-36x^2 + 25\end{align*}
  9. \begin{align*}4x^2 - y^2\end{align*}
  10. \begin{align*}16x^2 - 81y^2\end{align*}

Review (Answers)

To view the Review answers, open this PDF file and look for section 9.10. 

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